Abstract:Let $L(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L(H)$, the generalized derivation $\delta_{A,B}$ and the multiplication operator $M_{A,B}$ are defined on $L(H)$ by $\delta_{A,B}(X)=AX-XB$ and $M_{A,B}(X)=AXB$. In this paper, we give a characterization of bounded operators $A$ and $B$ such that the range of $M_{A,B}$ is closed. We present some sufficient conditions for $\delta_{A,B}$ to have closed range. Some related results are also given.
Keywords: generalized derivation; elementary operator; generalized inverse; Kato spectrum
DOI: DOI 10.14712/1213-7243.2025.004
AMS Subject Classification: 47A30 47A16 47B07 47B20 47B47